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A concept from the theory of almost-periodic functions (cf. [[Almost-periodic function|Almost-periodic function]]); a generalization of the notion of a period. For a uniformly almost-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a0119501.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a0119502.png" />, a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a0119503.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a0119505.png" />-almost-period of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a0119506.png" /> if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a0119507.png" />,
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A concept from the theory of almost-periodic functions (cf. [[Almost-periodic function|Almost-periodic function]]); a generalization of the notion of a period. For a uniformly almost-periodic function $f(x)$, $-\infty<x<\infty$, a number $\tau=\tau_f(\epsilon)$ is called an $\epsilon$-almost-period of $f(x)$ if for all $x$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a0119508.png" /></td> </tr></table>
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$$|f(x+\tau)-f(x)|<\epsilon.$$
  
For generalized almost-periodic functions the concept of an almost-period is more complicated. For example, in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a0119509.png" /> an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a01195010.png" />-almost-period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a01195011.png" /> is defined by the inequality
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For generalized almost-periodic functions the concept of an almost-period is more complicated. For example, in the space $S_l^p$ an $\epsilon$-almost-period $\tau$ is defined by the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a01195012.png" /></td> </tr></table>
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$$D_{S_l^p}[f(x+\tau),f(x)]<\epsilon,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a01195013.png" /> is the distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a01195014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a01195015.png" /> in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a01195016.png" />.
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where $D_{S_l^p}[f,\phi]$ is the distance between $f(x)$ and $\phi(x)$ in the metric of $S_l^p$.
  
A set of almost-periods of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a01195017.png" /> is said to be relatively dense if there is a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a01195018.png" /> such that every interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a01195019.png" /> of the real line contains at least one number from this set. The concepts of uniformly almost-periodic functions and that of Stepanov almost-periodic functions may be defined by requiring the existence of relatively-dense sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a01195020.png" />-almost-periods for these functions.
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A set of almost-periods of a function $f(x)$ is said to be relatively dense if there is a number $L=L(\epsilon,f)>0$ such that every interval $(\alpha,\alpha+L)$ of the real line contains at least one number from this set. The concepts of uniformly almost-periodic functions and that of Stepanov almost-periodic functions may be defined by requiring the existence of relatively-dense sets of $\epsilon$-almost-periods for these functions.
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
For the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a01195021.png" /> and its metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a01195022.png" /> see [[Almost-periodic function|Almost-periodic function]]. The Weyl, Besicovitch and Levitan almost-periodic functions can also be characterized in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a01195023.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011950/a01195024.png" />-periods. These characterizations are more complicated. A good additional reference is [[#References|[a1]]], especially Chapt. II.
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For the definition of $S_l^p$ and its metric $D_{S_l^p}$ see [[Almost-periodic function|Almost-periodic function]]. The Weyl, Besicovitch and Levitan almost-periodic functions can also be characterized in terms of $S_l^p$ $\epsilon$-periods. These characterizations are more complicated. A good additional reference is [[#References|[a1]]], especially Chapt. II.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.S. Besicovitch,  "Almost periodic functions" , Cambridge Univ. Press  (1932)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.S. Besicovitch,  "Almost periodic functions" , Cambridge Univ. Press  (1932)</TD></TR></table>

Latest revision as of 15:17, 18 July 2014

A concept from the theory of almost-periodic functions (cf. Almost-periodic function); a generalization of the notion of a period. For a uniformly almost-periodic function $f(x)$, $-\infty<x<\infty$, a number $\tau=\tau_f(\epsilon)$ is called an $\epsilon$-almost-period of $f(x)$ if for all $x$,

$$|f(x+\tau)-f(x)|<\epsilon.$$

For generalized almost-periodic functions the concept of an almost-period is more complicated. For example, in the space $S_l^p$ an $\epsilon$-almost-period $\tau$ is defined by the inequality

$$D_{S_l^p}[f(x+\tau),f(x)]<\epsilon,$$

where $D_{S_l^p}[f,\phi]$ is the distance between $f(x)$ and $\phi(x)$ in the metric of $S_l^p$.

A set of almost-periods of a function $f(x)$ is said to be relatively dense if there is a number $L=L(\epsilon,f)>0$ such that every interval $(\alpha,\alpha+L)$ of the real line contains at least one number from this set. The concepts of uniformly almost-periodic functions and that of Stepanov almost-periodic functions may be defined by requiring the existence of relatively-dense sets of $\epsilon$-almost-periods for these functions.

References

[1] B.M. Levitan, "Almost-periodic functions" , Moscow (1953) (In Russian)


Comments

For the definition of $S_l^p$ and its metric $D_{S_l^p}$ see Almost-periodic function. The Weyl, Besicovitch and Levitan almost-periodic functions can also be characterized in terms of $S_l^p$ $\epsilon$-periods. These characterizations are more complicated. A good additional reference is [a1], especially Chapt. II.

References

[a1] A.S. Besicovitch, "Almost periodic functions" , Cambridge Univ. Press (1932)
How to Cite This Entry:
Almost-period. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Almost-period&oldid=17082
This article was adapted from an original article by E.A. Bredikhina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article